NAG Library Function Document

nag_glopt_bnd_mcs_solve (e05jbc)

Note:this function usesoptional argumentsto define choices in the problem specification and in the details of the algorithm. If you wish to use default settings for all of the optional arguments, you need only read Sections 1 to 9 of this document. If, however, you wish to reset some or all of the settings please refer to Section 10 for a detailed description of the algorithm, and to Section 11 for a detailed description of the specification of the optional arguments.

1 Purpose

nag_glopt_bnd_mcs_solve (e05jbc) is designed to find the global minimum or maximum of an arbitrary function, subject to simple bound-constraints using a multi-level coordinate search method. Derivatives are not required, but convergence is only guaranteed if the objective function is continuous in a neighbourhood of a global optimum. It is not intended for large problems.

3 Description

nag_glopt_bnd_mcs_solve (e05jbc) is designed to solve modestly sized global optimization problems having simple bound-constraints only; it finds the global optimum of a nonlinear function subject to a set of bound constraints on the variables. Without loss of generality, the problem is assumed to be stated in the following form:

minimizex∈RnFx subject to ℓ≤x≤u and ℓ≤u,

where Fx (the objective function) is a nonlinear scalar function (assumed to be continuous in a neighbourhood of a global minimum), and the bound vectors are elements of R-n, where R- denotes the extended reals R∪-∞,∞. Relational operators between vectors are interpreted elementwise.

The optional argument Maximize should be set if you wish to solve maximization, rather than minimization, problems.

If certain bounds are not present, the associated elements of ℓ or u can be set to special values that will be treated as -∞ or +∞. See the description of the optional argument Infinite Bound Size. Phrases in this document containing terms like ‘unbounded values’ should be understood to be taken relative to this optional argument.

The initialization function nag_glopt_bnd_mcs_init (e05jac) does not need to be called before each invocation of nag_glopt_bnd_mcs_solve (e05jbc). You should be aware that a call to the initialization function will reset each optional argument to its default value, and, if you are using repeatable randomized initialization lists (see the description of the argument initmethod), the random state stored in
state
will be destroyed.

You must supply a function that evaluates Fx; derivatives are not required.

The method used by nag_glopt_bnd_mcs_solve (e05jbc) is based on MCS, the Multi-level Coordinate Search method described in Huyer and Neumaier (1999), and the algorithm it uses is described in detail in Section 10.

4 References

5 Arguments

Note: for convenience the subarray notation ai:j,k:l, as described in Section 3.2.1.4 in the Essential Introduction, is used. Using this notation, the term ‘column index’ refers to the index j in LISTi,j, say (see list for the definition of LIST).

1:
n – IntegerInput

On entry:
n, the number of variables.

Constraint:
n>0.

2:
objfun – function, supplied by the userExternal Function

objfun must evaluate the objective function Fx for a specified n-vector x.

On entry: x, the vector at which the objective function is to be evaluated.

3:
f – double *Output

On exit: must be set to the value of the objective function at x, unless you have specified termination of the current problem using inform.

4:
nstate – IntegerInput

On entry: if nstate=1 then nag_glopt_bnd_mcs_solve (e05jbc) is calling objfun for the first time. This argument setting allows you to save computation time if certain data must be read or calculated only once.

5:
comm – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to objfun.

user – double *

iuser – Integer *

p – Pointer

The type Pointer will be void *. Before calling nag_glopt_bnd_mcs_solve (e05jbc) you may allocate memory and initialize these pointers with various quantities for use by objfun when called from nag_glopt_bnd_mcs_solve (e05jbc) (see Section 3.2.1 in the Essential Introduction).

6:
inform – Integer *Output

On exit: must be set to a value describing the action to be taken by the solver on return from objfun. Specifically, if the value is negative the solution of the current problem will terminate immediately; otherwise, computations will continue.

3:
bound – Nag_BoundTypeInput

On entry: indicates whether the facility for dealing with bounds of special forms is to be used. bound must be set to one of the following values.

bound=Nag_Bounds

You will supply ℓ and u individually.

bound=Nag_NoBounds

There are no bounds on x.

bound=Nag_BoundsZero

There are semi-infinite bounds 0≤x.

bound=Nag_BoundsEqual

There are constant bounds ℓ=ℓ1 and u=u1.

Note that it only makes sense to fix any components of x when bound=Nag_Bounds.

For more information on methods initmethod=Nag_Linesearch, Nag_UserSet or Nag_Random see Section 10.1.

If ‘infinite’ values (as determined by the value of the optional argument Infinite Bound Size) are detected by nag_glopt_bnd_mcs_solve (e05jbc) when you are using a simple initialization method (initmethod=Nag_SimpleBdry or Nag_SimpleOffBdry), a safeguarded initialization procedure will be attempted, to avoid overflow.

On entry: bl is ℓ, the array of lower bounds. bu is u, the array of upper bounds.

If bound=Nag_Bounds, you must set
bl[i-1] to ℓi and bu[i-1] to ui, for i=1,2,…,n. If a particular xi is to be unbounded below, the corresponding bl[i-1] should be set to -infbnd, where infbnd is the value of the optional argument Infinite Bound Size. Similarly, if a particular xi is to be unbounded above, the corresponding bu[i-1] should be set to infbnd.

If bound=Nag_NoBounds or Nag_BoundsZero, arrays bl and bu need not be set on input.

If bound=Nag_BoundsEqual, you must set bl[0] to ℓ1 and bu[0] to u1. The remaining elements of bl and bu will then be populated by these initial values.

On entry:
must be set to,
at least, the maximum over i of the number of points in coordinate i at which to split according to the initialization list list; that is, sdlist≥maxinumpts[i-1].

Internally, nag_glopt_bnd_mcs_solve (e05jbc) uses list to determine sets of points along each coordinate direction to which it fits quadratic interpolants. Since fitting a quadratic requires at least three distinct points, this puts a lower bound on sdlist. Furthermore, in the case of initialization by linesearches (initmethod=Nag_Linesearch) internal storage considerations require that sdlist be at least 192.

Constraints:

if initmethod≠Nag_Linesearch, sdlist≥3;

if initmethod=Nag_Linesearch, sdlist≥192;

if initmethod=Nag_UserSet, sdlist≥maxinumpts[i-1].

8:
list[n×sdlist] – doubleInput/Output

Note: where LISTi,j appears in this document, it refers to the array element
list[i-1×sdlist+j-1].

For convenience the subarray notation LISTi:j,k:l, as described in Section 3.2.1.4 in the Essential Introduction, is used. Using this notation, the term ‘column index’ refers to the index j in LISTi,j, say.

On entry: this argument need not be set on entry if you wish to use one of the preset initialization methods (initmethod≠Nag_UserSet).

list is the ‘initialization list’: whenever a sub-box in the algorithm is split for the first time (either during the initialization procedure or later), for each non-fixed coordinate i the split is done at the values LISTi,1:numpts[i-1], as well as at some adaptively chosen intermediate points. The array sections
LISTi,1:numpts[i-1], for i=1,2,…,n, must be in ascending order with each entry being distinct. In this context, ‘distinct’ should be taken to mean relative to the safe-range argument (see nag_real_safe_small_number (X02AMC)).

On entry: this argument need not be set on entry if you wish to use one of the preset initialization methods (initmethod≠Nag_UserSet).

You must designate a point stored in list that you wish nag_glopt_bnd_mcs_solve (e05jbc) to consider as an ‘initial point’ for the purposes of the splitting procedure. Call this initial point x*. The coordinates of x* correspond to a set of indices
Ji, for i=1,2,…,n, such that xi* is stored in
LISTi,Ji, for i=1,2,…,n. You must set
initpt[i-1]=Ji, for i=1,2,…,n.

Constraint:
if x[i-1] is not fixed, 1≤initpt[i-1]≤sdlist, for i=1,2,…,n.

11:
monit – function, supplied by the userExternal Function

monit may be used to monitor the optimization process. It is invoked upon every successful completion of the procedure in which a sub-box is considered for splitting. It will also be called just before nag_glopt_bnd_mcs_solve (e05jbc) exits if that splitting procedure was not successful.

Pointer to structure of type Nag_Comm; the following members are relevant to monit.

user – double *

iuser – Integer *

p – Pointer

The type Pointer will be void *. Before calling nag_glopt_bnd_mcs_solve (e05jbc) you may allocate memory and initialize these pointers with various quantities for use by monit when called from nag_glopt_bnd_mcs_solve (e05jbc) (see Section 3.2.1 in the Essential Introduction).

15:
inform – Integer *Output

On exit: must be set to a value describing the action to be taken by the solver on return from monit. Specifically, if the value is negative the solution of the current problem will terminate immediately; otherwise, computations will continue.

nag_glopt_bnd_mcs_solve (e05jbc) returns with fail.code=NE_NOERROR if your termination criterion has been met: either a target value has been found to the required relative error (as determined by the values of the optional arguments Target Objective Value, Target Objective Error and Target Objective Safeguard), or the best function value was static for the number of sweeps through levels given by the optional argument Static Limit. The latter criterion is the default.

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument value had an illegal value.

NE_DIV_COMPLETE

The division procedure completed but your target value
could not be reached.
Despite every sub-box being processed Splits Limit times,
the target value you provided in Target Objective Value could not
be found to the tolerances given in Target Objective Error and
Target Objective Safeguard. You could try reducing Splits Limit or the objective tolerances.

NE_INF_INIT_LIST

A finite initialization list could not be computed internally.
Consider reformulating the bounds on the problem,
try providing your own initialization list, use the randomization option
(initmethod=Nag_Random) or vary the value of
Infinite Bound Size.

An error occurred during initialization.
It is likely that points from the initialization list are
very close together. Try relaxing the bounds on the variables or
use a different initialization method.

NE_INT

On entry, initmethod=Nag_Linesearch and sdlist=value.
Constraint: if initmethod=Nag_Linesearch then sdlist≥192.

On entry, initmethod=value and sdlist=value.
Constraint: if initmethod≠Nag_Linesearch then sdlist≥3.

A value of Splits Limit (smax) smaller than nr+3 was set:
smax=value, nr=value.

On entry, user-supplied initpt[i-1]=value, i=value.
Constraint: if x[i-1] is not fixed then
initpt[i-1]≥1, for i=1,2,…,n.

On entry, user-supplied initpt[i-1]=value,
i=value and sdlist=value.
Constraint: if x[i-1] is not fixed then
initpt[i-1]≤sdlist, for i=1,2,…,n.

On entry, user-supplied numpts[i-1]=value, i=value.
Constraint: if x[i-1] is not fixed then
numpts[i-1]≥3, for i=1,2,…,n.

On entry, user-supplied numpts[i-1]=value,
i=value and sdlist=value.
Constraint: if x[i-1] is not fixed then
numpts[i-1]≤sdlist, for i=1,2,…,n.

On entry, user-supplied section LISTi,1:numpts[i-1] was not in ascending order: numpts[i-1]=value, i=value.

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

NE_LINESEARCH_ERROR

An error occurred during linesearching.
It is likely that your objective function is badly scaled: try rescaling it.
Also, try relaxing the bounds or use a different initialization method.
If the problem persists, please contact NAG quoting error code value.

On entry, bound=Nag_BoundsEqual and bl[0]=bu[0]=value.
Constraint: if bound=Nag_BoundsEqual then bl[0]<bu[0].

NE_REAL_2

On entry, bound=Nag_Bounds or Nag_BoundsEqual and bl[i-1]=value,
bu[i-1]=value and i=value.
Constraint: if bound=Nag_Bounds then
bl[i-1]≤bu[i-1], for i=1,2,…,n;
if bound=Nag_BoundsEqual then bl[0]<bu[0].

On entry, user-supplied LISTi,j=value,
i=value, j=value, and bl[i-1]=value.
Constraint: if x[i-1] is not fixed then
LISTi,j≥bl[i-1], for i=1,2,…,n and j=1,2,…,numpts[i-1].

On entry, user-supplied LISTi,j=value,
i=value, j=value, and bu[i-1]=value.
Constraint: if x[i-1] is not fixed then
LISTi,j≤bu[i-1], for i=1,2,…,n and j=1,2,…,numpts[i-1].

NE_TOO_MANY_FEVALS

The function evaluations limit was exceeded.
Approximately Function Evaluations Limit function calls
have been made without your chosen termination criterion
being satisfied.

7 Accuracy

If fail.code=NE_NOERROR on exit, then the vector returned in the array x is an estimate of the solution x whose function value satisfies your termination criterion: the function value was static for Static Limit sweeps through levels, or

Fx-objval≤maxobjerr×objval,objsfg,

where objval is the value of the optional argument Target Objective Value, objerr is the value of the optional argument Target Objective Error, and objsfg is the value of the optional argument Target Objective Safeguard.

8 Further Comments

For each invocation of nag_glopt_bnd_mcs_solve (e05jbc), local workspace arrays of fixed length are allocated internally. The total size of these arrays amounts to 13nr+smax-1 Integer elements, where smax is the value of the optional argument Splits Limit and nr is the number of non-fixed variables, and 2+nrsdlist+2n+21nr+3nr2+1 double elements. In addition, if you are using randomized initialization lists (see the description of the argument initmethod), a further 21 Integer elements are allocated internally.

In order to keep track of the regions of the search space that have been visited while looking for a global optimum, nag_glopt_bnd_mcs_solve (e05jbc) internally allocates arrays of increasing sizes depending on the difficulty of the problem. Two of the main factors that govern the amount allocated are the number of sub-boxes (call this quantity nboxes) and the number of points in the ‘shopping basket’ (the argument nbaskt on entry to monit). Safe, pessimistic upper bounds on these two quantities are so large as to be impractical. In fact, the worst-case number of sub-boxes for even the most simple initialization list (when ninit=3 on entry to monit) grows like nrnr. Thus nag_glopt_bnd_mcs_solve (e05jbc) does not attempt to estimate in advance the final values of nboxes or nbaskt for a given problem. There are a total of 5 Integer arrays and 4+nr+ninit double arrays whose lengths depend on nboxes, and there are a total of 2 Integer arrays and 3+n+nr double arrays whose lengths depend on nbaskt. nag_glopt_bnd_mcs_solve (e05jbc) makes a fixed initial guess that the maximum number of sub-boxes required will be 10000 and that the maximum number of points in the ‘shopping basket’ will be 1000. If ever a greater amount of sub-boxes or more room in the ‘shopping basket’ is required, nag_glopt_bnd_mcs_solve (e05jbc) performs reallocation, usually doubling the size of the inadequately-sized arrays. Clearly this process requires periods where the original array and its extension exist in memory simultaneously, so that the data within can be copied, which compounds the complexity of nag_glopt_bnd_mcs_solve (e05jbc)'s memory usage. It is possible (although not likely) that if your problem is particularly difficult to solve, or of a large size (hundreds of variables), you may run out of memory.

One array that could be dynamically resized by nag_glopt_bnd_mcs_solve (e05jbc) is the ‘shopping basket’ (xbaskt on entry to monit). If the initial attempt to allocate 1000nrdoubles for this array fails, monit will not be called on exit from nag_glopt_bnd_mcs_solve (e05jbc).

nag_glopt_bnd_mcs_solve (e05jbc) performs better if your problem is well-scaled. It is worth trying (by guesswork perhaps) to rescale the problem if necessary, as sensible scaling will reduce the difficulty of the optimization problem, so that nag_glopt_bnd_mcs_solve (e05jbc) will take less computer time.

9 Example

This example finds the global minimum of the ‘peaks’ function in two dimensions

Fx,y=31-x2exp-x2-y+12-10x5-x3-y5exp-x2-y2-13exp-x+12-y2

on the box -3,3×-3,3.

The function F has several local minima and one global minimum in the given box. The global minimum is approximately located at 0.23,-1.63, where the function value is approximately -6.55.

We use default values for all the optional arguments, and we instruct nag_glopt_bnd_mcs_solve (e05jbc) to use the simple initialization list corresponding to initmethod=Nag_SimpleBdry. In particular, this will set for us the initial point 0,0 (see Section 9.3).

10 Algorithmic Details

Here we summarize the main features of the MCS algorithm used in nag_glopt_bnd_mcs_solve (e05jbc), and we introduce some terminology used in the description of the function and its arguments. We assume throughout that we will only do any work in coordinates i in which xi is free to vary. The MCS algorithm is fully described in Huyer and Neumaier (1999).

10.1 Initialization and Sweeps

Each sub-box is determined by a basepoint x and an opposite pointy. We denote such a sub-box by Bx,y. The basepoint is allowed to belong to more than one sub-box, is usually a boundary point, and is often a vertex.

An initialization procedure produces an initial set of sub-boxes. Whenever a sub-box is split along a coordinate i for the first time (in the initialization procedure or later), the splitting is done at three or more user-defined values xijj at which the objective function is sampled, and at some adaptively chosen intermediate points. At least four children are generated. More precisely, we assume that we are given

ℓi≤xi1<xi2<⋯<xiLi≤ui, Li≥3, for ​i=1,2,…,n

and a vector p that, for each i, locates within xijj the ith coordinate of an initial pointx0; that is, if xi0=xij for some j=1,2,…,Li, then pi=j. A good guess for the global optimum can be used as x0.

The initialization points and the vectors ℓ and p are collectively called the initialization list (and sometimes we will refer to just the initialization points as ‘the initialization list’, whenever this causes no confusion). The initialization data may be input by you, or they can be set to sensible default values by nag_glopt_bnd_mcs_solve (e05jbc): if you provide them yourself, LISTi,j should contain xij, numpts[i-1] should contain Li, and initpt[i-1] should contain
pi, for i=1,2,…,n and j=1,2,…,Li; if you wish nag_glopt_bnd_mcs_solve (e05jbc) to use one of its preset initialization methods, you could choose one of two simple, three-point methods (see Figure 1). If the list generated by one of these methods contains infinite values, attempts are made to generate a safeguarded list using the function subintx,y (which is also used during the splitting procedure, and is described in Section 10.2). If infinite values persist, nag_glopt_bnd_mcs_solve (e05jbc) exits with fail.code=NE_INF_INIT_LIST. There is also the option to generate an initialization list with the aid of linesearches (by setting initmethod=Nag_Linesearch). Starting with the absolutely smallest point in the root box, linesearches are made along each coordinate. For each coordinate, the local minimizers found by the linesearches are put into the initialization list. If there were fewer than three minimizers, they are augmented by close-by values. The final preset initialization option (initmethod=Nag_Random) generates a randomized list, so that independent multiple runs may be made if you suspect a global optimum has not been found. Each call to the initialization function nag_glopt_bnd_mcs_init (e05jac) resets the initial-state vector for the Wichmann–Hill base-generator that is used. Depending on whether you set the optional argument Repeatability to ‘ON’ or ‘OFF’, the random state is initialized to give a repeatable or non-repeatable sequence. Then, a random integer between 3 and sdlist is selected, which is then used to determine the number of points to be generated in each coordinate; that is, numpts becomes a constant vector, set to this value. The components of list are then generated, from a uniform distribution on the root box if the box is finite, or else in a safeguarded fashion if any bound is infinite. The array initpt is set to point to the best point in list.

Given an initialization list (preset or otherwise), nag_glopt_bnd_mcs_solve (e05jbc) evaluates F at x0, and sets the initial estimate of the global minimum, x*, to x0. Then, for i=1,2,…,n, the objective function F is evaluated at Li-1 points that agree with x* in all but the ith coordinate. We obtain pairs
x^j,fij, for j=1,2,…,Li, with: x*=x^j1, say; with, for j≠j1,

x^kj=xk*if ​k≠i;xkjotherwise;

and with

fij=Fx^j.

The point having the smallest function value is renamed x* and the procedure is repeated with the next coordinate.

Once nag_glopt_bnd_mcs_solve (e05jbc) has a full set of initialization points and function values, it can generate an initial set of sub-boxes. Recall that the root box is Bx,y=ℓ,u, having basepoint x=x0. The opposite point y is a corner of ℓ,u farthest away from x, in some sense. The point x need not be a vertex of ℓ,u, and y is entitled to have infinite coordinates. We loop over each coordinate i, splitting the current box along coordinate i into 2Li-2, 2Li-1 or 2Li subintervals with exactly one of the x^ij as endpoints, depending on whether two, one or none of the x^ij are on the boundary. Thus, as well as splitting at
x^ij, for j=1,2,…,Li, we split at additional points
zij, for j=2,3,…,Li. These additional zij are such that

zij=x^ij-1+qmx^ij-x^ij-1, j=2,…,Li,

where q is the golden-section ratio 5-1/2, and the exponent m takes the value 1 or 2, chosen so that the sub-box with the smaller function value gets the larger fraction of the interval. Each child sub-box gets as basepoint the point obtained from x* by changing xi* to the xij that is a boundary point of the corresponding ith coordinate interval; this new basepoint therefore has function value fij. The opposite point is derived from y by changing yi to the other end of that interval.

nag_glopt_bnd_mcs_solve (e05jbc) can now rank the coordinates based on an estimated variability of F. For each i we compute the union of the ranges of the quadratic interpolant through any three consecutive x^ij, taking the difference between the upper and lower bounds obtained as a measure of the variability of F in coordinate i. A vector π is populated in such a way that coordinate i has the πith highest estimated variability. For tiebreaks, when the x* obtained after splitting coordinate i belongs to two sub-boxes, the one that contains the minimizer of the quadratic models is designated the current sub-box for coordinate i+1.

Boxes are assigned levels in the following manner. The root box is given level 1. When a sub-box of level s is split, the child with the smaller fraction of the golden-section split receives level s+2; all other children receive level s+1. The box with the better function value is given the larger fraction of the splitting interval and the smaller level because then it is more likely to be split again more quickly. We see that after the initialization procedure the first level is empty and the non-split boxes have levels 2,…,nr+2, so it is meaningful to choose smax much larger than nr. Note that the internal structure of nag_glopt_bnd_mcs_solve (e05jbc) demands that smax be at least nr+3.

Examples of initializations in two dimensions are given in Figure 1. In both cases the initial point is x0=ℓ+u/2; on the left the initialization points are

x1=ℓ, x2=ℓ+u/2, x3=u,

while on the right the points are

x1=5ℓ+u/6, x2=ℓ+u/2, x3=ℓ+5u/6.

In Figure 1, basepoints and levels after initialization are displayed. Note that these initialization lists correspond to initmethod=Nag_SimpleBdry and initmethod=Nag_SimpleOffBdry, respectively.

Figure 1: Examples of the initialization procedure

After initialization, a series of sweeps through levels is begun. A sweep is defined by three steps:

(i)

scan the list of non-split sub-boxes. Fill a record listb according to bs=0 if there is no box at level s, and with bs pointing to a sub-box with the lowest function value among all sub-boxes with level s otherwise, for 0<s<smax;

(ii)

the sub-box with label bs is a candidate for splitting. If the sub-box is not to be split, according to the rules described in Section 10.2, increase its level by 1 and update bs+1 if necessary. If the sub-box is split, mark it so, insert its children into the list of sub-boxes, and update b if any child with level s′ yields a strict improvement of F over those sub-boxes at level s′;

(iii)

increment s by 1. If s=smax then displaying monitoring information and start a new sweep; else if bs=0 then repeat this step; else display monitoring information and go to the previous step.

Clearly, each sweep ends after at most smax-1 visits of the third step.

10.2 Splitting

Each sub-box is stored by nag_glopt_bnd_mcs_solve (e05jbc) as a set of information about the history of the sub-box: the label of its parent, a label identifying which child of the parent it is, etc. Whenever a sub-box Bx,y of level s<smax is a candidate for splitting, as described in Section 10.1, we recover x, y, and the number, nj, of times coordinate j has been split in the history of B. Sub-box B could be split in one of two ways.

(i)

Splitting by rank

If s>2nrmin⁡nj+1, the box is always split. The splitting index is set to a coordinate i such that ni=min⁡nj.

(ii)

Splitting by expected gain

If s≤2nrmin⁡nj+1, the sub-box could be split along a coordinate where a maximal gain in function value is expected. This gain is estimated according to a local separable quadratic model obtained by fitting to 2nr+1 function values. If the expected gain is too small the sub-box is not split at all, and its level is increased by 1.

Eventually, a sub-box that is not eligible for splitting by expected gain will reach level 2nrmin⁡nj+1+1 and then be split by rank, as long as smax is large enough. As smax→∞, the rule for splitting by rank ensures that each coordinate is split arbitrarily often.

Before describing the details of each splitting method, we introduce the procedure for correctly handling splitting at adaptive points and for dealing with unbounded intervals. Suppose we want to split the ith coordinate interval ▯xi,yi, where we define ▯xi,yi=minxi,yi,maxxi,yi, for xi∈R and yi∈R-, and where x is the basepoint of the sub-box being considered. The descendants of the sub-box should shrink sufficiently fast, so we should not split too close to xi. Moreover, if yi is large we want the new splitting value to not be too large, so we force it to belong to some smaller interval ▯ξ′,ξ′′, determined by

10.2.1 Splitting by rank

Consider a sub-box B with level s>2nrmin⁡nj+1. Although the sub-box has reached a high level, there is at least one coordinate along which it has not been split very often. Among the i such that ni=min⁡nj for B, select the splitting index to be the coordinate with the lowest πi (and hence highest variability rank). ‘Splitting by rank’ refers to the ranking of the coordinates by ni and πi.

If ni=0, so that B has never been split along coordinate i, the splitting is done according to the initialization list and the adaptively chosen golden-section split points, as described in Section 10.1. Also as covered there, new basepoints and opposite points are generated. The children having the smaller fraction of the golden-section split (that is, those with larger function values) are given level mins+2,smax. All other children are given level s+1.

Otherwise, B ranges between xi and yi in the ith coordinate direction. The splitting value is selected to be zi=xi+2subintxi,yi-xi/3; we are not attempting to split based on a large reduction in function value, merely in order to reduce the size of a large interval, so zi may not be optimal. Sub-box B is split at zi and the golden-section split point, producing three parts and requiring only one additional function evaluation, at the point x′ obtained from x by changing the ith coordinate to zi. The child with the smaller fraction of the golden-section split is given level mins+2,smax, while the other two parts are given level s+1. Basepoints are assigned as follows: the basepoint of the first child is taken to be x, and the basepoint of the second and third children is the point x′. Opposite points are obtained by changing yi to the other end of the ith coordinate-interval of the corresponding child.

10.2.2 Splitting by expected gain

When a sub-box B has level s≤2nrmin⁡nj+1, we compute the optimal splitting index and splitting value from a local separable quadratic used as a simple local approximation of the objective function. To fit this curve, for each coordinate we need two additional points and their function values. Such data may be recoverable from the history of B: whenever the ith coordinate was split in the history of B, we obtained values that can be used for the current quadratic interpolation in coordinate i.

We loop over i; for each coordinate we pursue the history of B back to the root box, and we take the first two points and function values we find, since these are expected to be closest to the current basepoint x. If the current coordinate has not yet been split we use the initialization list. Then we generate a local separable model eξ for Fξ by interpolation at x and the 2nr additional points just collected:

eξ=Fx+∑i=1neiξi.

We define the expected gaine^i in function value when we evaluate at a new point obtained by changing coordinate i in the basepoint, for each i, based on two cases:

(i)

ni=0. We compute the expected gain as

e^i=min1≤j≤Lifij-fipi.

Again, we split according to the initialization list, with the new basepoints and opposite points being as before.

(ii)

ni>0. Now, the ith component of our sub-box ranges from xi to yi. Using the quadratic partial correction function

eiξi=αiξi-xi+βiξi-xi2

we can approximate the maximal gain expected when changing xi only. We will choose the splitting value from ▯ξ′,ξ′′. We compute

e^i=minξi∈▯ξ′,ξ′′eiξi

and call zi the minimizer in ▯ξ′,ξ′′.

If the expected best function value fexp satisfies

fexp=Fx+min1≤i≤ne^i<fbest,

(1)

where fbest is the current best function value (including those function values obtained by local optimization), we expect the sub-box to contain a better point and so we split it, using as splitting index the component with minimal e^i. Equation (1) prevents wasting function calls by avoiding splitting sub-boxes whose basepoints have bad function values. These sub-boxes will eventually be split by rank anyway.

We now have a splitting index and a splitting value zi. The sub-box is split at zi as long as zi≠yi, and at the golden-section split point; two or three children are produced. The larger fraction of the golden-section split receives level s+1, while the smaller fraction receives level mins+2,smax. If it is the case that zi≠yi and the third child is larger than the smaller of the two children from the golden-section split, the third child receives level s+1. Otherwise it is given the level mins+2,smax. The basepoint of the first child is set to x, and the basepoint of the second (and third if it exists) is obtained by changing the ith coordinate of x to zi. The opposite points are again derived by changing yi to the other end of the ith coordinate interval of B.

If equation (1) does not hold, we expect no improvement. We do not split, and we increase the level of B by 1.

10.3 Local Search

The local optimization algorithm used by nag_glopt_bnd_mcs_solve (e05jbc) uses linesearches along directions that are determined by minimizing quadratic models, all subject to bound constraints. Triples of vectors are computed using coordinate searches based on linesearches. These triples are used in triple search procedures to build local quadratic models for F. A trust-region-type approach to minimize these models is then carried out, and more information about the coordinate search and the triple search can be found in Huyer and Neumaier (1999).

The local search starts by looking for better points without being too local, by making a triple search using points found by a coordinate search. This yields a new point and function value, an approximation of the gradient of the objective, and an approximation of the Hessian of the objective. Then the quadratic model for F is minimized over a small box, with the solution to that minimization problem then being used as a linesearch direction to minimize the objective. A measure r is computed to quantify the predictive quality of the quadratic model.

The third stage is the checking of termination criteria. The local search will stop if more than loclim visits to this part of the local search have occurred, where loclim is the value of the optional argument Local Searches Limit. If that is not the case, it will stop if the limit on function calls has been exceeded (see the description of the optional argument Function Evaluations Limit). The final criterion checks if no improvement can be made to the function value, or whether the approximated gradient g is small, in the sense that

gTmaxx,xold<loctolf0-f.

The vector xold is the best point at the start of the current loop in this iterative local-search procedure, the constant loctol is the value of the optional argument Local Searches Tolerance, f is the objective value at x, and f0 is the smallest function value found by the initialization procedure.

Next, nag_glopt_bnd_mcs_solve (e05jbc) attempts to move away from the boundary, if any components of the current point lie there, using linesearches along the offending coordinates. Local searches are terminated if no improvement could be made.

The fifth stage carries out another triple search, but this time it does not use points from a coordinate search, rather points lying within the trust-region box are taken.

The final stage modifies the trust-region box to be bigger or smaller, depending on the quality of the quadratic model, minimizes the new quadratic model on that box, and does a linesearch in the direction of the minimizer. The value of r is updated using the new data, and then we go back to the third stage (checking of termination criteria).

The Hessians of the quadratic models generated by the local search may not be positive definite, so nag_glopt_bnd_mcs_solve (e05jbc) uses the general nonlinear optimizer nag_opt_sparse_nlp_solve (e04vhc) to minimize the models.

11 Optional Arguments

Several optional arguments in nag_glopt_bnd_mcs_solve (e05jbc) define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of nag_glopt_bnd_mcs_solve (e05jbc) these optional arguments have associated default values that are appropriate for most problems. Therefore, you need only specify those optional arguments whose values are to be different from their default values.

The remainder of this section can be skipped if you wish to use the default values for all optional arguments. The following is a list of the optional arguments available and a full description of each optional argument is provided in Section 11.1.

nag_glopt_bnd_mcs_optset_file (e05jcc) reads options from an external options file, with Begin and End as the first and last lines respectively, and with each intermediate line defining a single optional argument. For example,

All optional arguments not specified by you are set to their default values. Valid values of optional arguments specified by you are unaltered by nag_glopt_bnd_mcs_solve (e05jbc) and so remain in effect for subsequent calls to nag_glopt_bnd_mcs_solve (e05jbc), unless you explicitly change them.

11.1 Description of the Optional Arguments

For each option, we give a summary line, a description of the optional argument and details of constraints.

The summary line contains:

a parameter value,
where the letters a, i​ and ​r denote options that take character, integer and real values respectively, and where the letter a denotes an option that takes an ‘ON’ or ‘OFF’ value;

Option names are case-insensitive and must be provided in full; abbreviations are not recognized.

Defaults

This special keyword is used to reset all optional arguments to their default values, and any random state stored in
state
will be destroyed.

Any option value given with this keyword will be ignored. This optional argument cannot be queried or got.

Function Evaluations Limit

i

Default =100nr2

This puts an approximate limit on the number of function calls allowed. The total number of calls made is checked at the top of an internal iteration loop, so it is possible that a few calls more than nf may be made.

Constraint:
nf>0.

Infinite Bound Size

r

Default =rmax14

This defines the ‘infinite’ bound infbnd in the definition of the problem constraints. Any upper bound greater than or equal to infbnd will be regarded as ∞ (and similarly any lower bound less than or equal to -infbnd will be regarded as -∞).

Constraint:
rmax14≤infbnd≤rmax12.

Local Searches

a

Default ='ON'

If you want to try to accelerate convergence of nag_glopt_bnd_mcs_solve (e05jbc) by starting local searches from candidate minima, you will require lcsrch to be ‘ON’.

Constraint:
lcsrch='ON'​ or ​'OFF'.

Local Searches Limit

i

Default =50

This defines the maximal number of iterations to be used in the trust-region loop of the local-search procedure.

Constraint:
loclim>0.

Local Searches Tolerance

r

Default =2ε

The value of loctol is the multiplier used during local searches as a stopping criterion for when the approximated gradient is small, in the sense described in Section 10.3.

Constraint:
loctol≥2ε.

Minimize

Default

Maximize

These keywords specify the required direction of optimization. Any option value given with these keywords will be ignored.

Nolist

Default

List

These options control the echoing of each optional argument specification as it is supplied. List turns printing on, Nolist turns printing off. The output is sent to
stdout.

Any option value given with these keywords will be ignored. This optional argument cannot be queried or got.

Repeatability

a

Default ='OFF'

For use with random initialization lists (initmethod=Nag_Random). When set to ‘ON’, an internally-initialized random state is stored in
state
for use in subsequent calls to nag_glopt_bnd_mcs_solve (e05jbc).

Constraint:
repeat='ON'​ or ​'OFF'.

Splits Limit

i

Default =dnr+2/3

Along with the initialization list list, this defines a limit on the number of times the root box will be split along any single coordinate direction. If Local Searches is ‘OFF’ you may find the default value to be too small.

Constraint:
smax>nr+2.

Static Limit

i

Default =3nr

As the default termination criterion, computation stops when the best function value is static for stclim sweeps through levels. This argument is ignored if you have specified a target value to reach in Target Objective Value.

Constraint:
stclim>0.

Target Objective Error

r

Default =ε14

If you have given a target objective value to reach in objval (the value of the optional argument Target Objective Value), objerr sets your desired relative error (from above if Minimize is set, from below if Maximize is set) between obj and objval, as described in Section 7. See also the description of the optional argument Target Objective Safeguard.

Constraint:
objerr≥2ε.

Target Objective Safeguard

r

Default =ε12

If you have given a target objective value to reach in objval (the value of the optional argument Target Objective Value), objsfg sets your desired safeguarded termination tolerance, for when objval is close to zero.

Constraint:
objsfg≥2ε.

Target Objective Value

r

This argument may be set if you wish nag_glopt_bnd_mcs_solve (e05jbc) to use a specific value as the target function value to reach during the optimization. Setting objval overrides the default termination criterion determined by the optional argument Static Limit.